Classical model of the upper bounds of the cascading contribution to the second hyperpolarizability

We investigate whether microscopic cascading of second-order nonlinearities of two molecules in the side-by-side configuration can lead to a third-order molecular nonlinear-optical response that exceeds the fundamental limit. We find that for large values of the second hyperpolarizability, the side-by-side configuration has a cascading contribution that lowers the direct contribution. However, we do find that there is a cascading contribution to the second hyperpolarizability when there is no direct contribution. Thus, while cascading can never lead to a larger nonlinear-optical response than for a single molecule with the same number of electrons, it may provide design flexibility in making large third-order susceptibility materials when the molecular second hyperpolarizability vanishes.

Figure 1

(a) Third-harmonic generation from two second-order processes. (b) Phase-matched third-harmonic generation in a two step process. (c) A Feynman diagram for one configuration of microscopic cascading. (d) The direct third-order process of third-harmonic generation.

Figure 2

Two molecules are aligned in a static electric field, $\vec{E}$, and are separated by a distance $r$.

Figure 3

Two molecules in a static electric field, $\vec{E}$, are separated by a distance $r$ and are in the counteraligned geometry.

Figure 4

(a) ${\alpha }^{\mathrm{int}}$, (b) ${\beta }^{\mathrm{int}}$, and (c) ${\gamma }^{\mathrm{int}}$ as functions of $E$ and $X$.

Figure 5

The effective molecular susceptibilities ${\alpha }_{\mathrm{eff}}(X,E,r)$, ${\beta }_{\mathrm{eff}}(X,E,r)$, and ${\gamma }_{\mathrm{eff}}(X,E,r)$, when the distance of separation is $r^{\prime }=1$ and $r^{\prime }=10$.

Figure 6

The function $X\left(E\right)$ when $\gamma =0$ for $0⩽X⩽1$ and $0⩽E⩽1$. The insets show an expanded view.

Figure 7

(a) A surface plot of ${\gamma }_{\mathrm{eff}}^{\mathrm{int}}(r^{\prime },E)$ along the branch corresponding to the solid blue curve in Fig. 6. (b) ${\gamma }_{\mathrm{eff}}^{\mathrm{int}}(r^{\prime },X)$, along the dashed red curve.